Question: Dirichlet and multinomial distributions. As a generalisation of Example (7.36), consider an urn model in which each ball has one of a finite number s

Dirichlet and multinomial distributions. As a generalisation of Example (7.36), consider an urn model in which each ball has one of a finite number s of colours (instead of only two).

Let ‚ be the set of all probability densities on ¹1; : : : ; sº. Suppose that the prior distribution

˛ on ‚ is the Dirichlet distribution D for some parameter 2 0;1OEs , which is defined by the equation

DP(A) = (14) 9-18, ... s-1, A Bo (1 (1) ((1) i=1

Here, the integral runs over all .#1; : : : ; #s1/ for which # WD .#1; : : : ; #s1; 1
Ps1 iD1 #i /
belongs to ‚. The fact that D is indeed a probability measure will follow for instance from Problem 9.8. (In the case 1, D is the uniform distribution on ‚.) We take a sample of size n with replacement. For each colour composition # 2 ‚, the colour histogram then has the multinomial distributionMn;#. Determine the associated posterior distribution.

DP(A) = (14) 9-18, ... s-1, A Bo (1 (1) ((1) i=1

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